### 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考| Scenario Generation

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Scenario Generation

The optimal ALM strategy in the model depends on an extended set of asset and liability processes. We assume a multivariate normal distribution for monthly price returns of a representative set of market benchmarks including fixed income, equity, and real estate indices. The following $I=12$ investment opportunities are considered: for $i=1$ : the EURIBOR 3 months; for $i=2,3,4,5$, 6: the Barclays Treasury indices for maturity buckets $1-3,3-5,5-7,7-10$, and $10+$ years, respectively; for $i=7,8,9,10$ : the Barclays Corporate indices, again spanning maturities $1-3,3-5,5-7$, and $7-10$ years; finally for $i=11$ : the GPR Real Estate Europe index and $i=12$ : the MSCI EMU equity index. Recall that in our notation $I_{1}$ includes all fixed income assets and $I_{2}$ the real estate and equity investments.

The benchmarks represent total return indices incorporating over time the securities’ cash payments. In the definition of the strategic asset allocation problem, unlike
5 Dynamic Portfolio Management for Property and Casualty Insuranee
109
real estate and equity investments, money market and fixed income benchmarks are assumed to have a fixed maturity equal to their average duration ( 3 months, 2 years, 4,6 , and $8.5$ years for the corresponding Treasury and Corporate indices and 12 years for the $10+$ Treasury index). Income cashflows due to coupon payments will be estimated ex post through an approximation described below upon selling or expiry of fixed income benchmarks and disentangled from price returns. Equity investments will instead generate annual dividends through a price-adjusted random dividend yield. Finally, for real estate investments a simple model focusing only on price returns is considered. Scenario generation translates the indices return trajectories for the above market benchmarks into a tree process (see, for instance, the tree structure in Fig. 5.3) for the ALM coefficients (Consigli et al., 2010; Chapters 15 and 16 ): this is referred to in the sequel as the tree coefficient process. We rely on the process nodal realizations (Chapter 4 ) to identify the coefficients to be associated with each decision variable in the ALM model. We distinguish in the decision model between asset and liability coefficients.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Asset Return Scenarios

The following random coefficients must be derived from the data process simulations along the specified scenario tree. For each scenario $s$, at $t=t_{0}, t_{1}, \ldots, t_{n}=H$, we define
$\rho_{i, t}(s)$ : the return of asset $i$ at time $t$ in scenario $s$;
$\delta_{i, t}(s)$ : the dividend yield of asset $i$ at time $t$ in scenario $s$;
$\eta_{i, h, t}(s)$ : a positive interest at time $t$ in scenario $s$ per unit investment in asset $i \in I_{1}$, in epoch $h$;
$\gamma_{i, h, t}(s)$ : the capital gain at time $t$ in scenario $s$ per unit investment in asset $i \in I_{2}$ in epoch $h$;
$\zeta_{t}^{+/-}(s):$ the (positive and negative) interest rates on the cash account at time $t$ in scenario $s$.

Price refurns $\rho_{i, t}(s)$ under scenario s are directly computed from the associated benchmarks $V_{i, t}(s)$ assuming a multivariate normal distribution, with $\rho:=$ $\left{\rho_{i}(\omega)\right}, \rho \sim N(\mu, \Sigma)$, where $\omega$ denotes a generic random element, and $\mu$ and $\Sigma$ denote the return mean vector and variance-covariance matrix, respectively. In the statistical model we consider for $i \in I 12$ investment opportunities and the inflation rate. Denoting by $\Delta t$ a monthly time increment and given the initial values $V_{i, 0}$ for $i=1,2, \ldots, 12$, we assume a stochastic difference equation for $V_{i, t}:$
\begin{aligned} \rho_{i, t}(s) &=\frac{V_{i, t}(s)-V_{i, t-\Delta t}(s)}{V_{i, t}-\Delta t(s)}, \ \frac{\Delta V_{i, t}(s)}{V_{i, t}(s)}=\mu \Delta t+\Sigma \Delta W_{t}(s) . \end{aligned}

In $(5.12), \Delta W_{t} \sim N(0, \Delta t)$. We show in Tables $5.2$ and $5.3$ the estimated statistical parameters adopted to generate through Monte Carlo simulation (Consigli et al., 2010 ; Glasserman, 2003 ) the correlated monthly returns in (5.12) for each benchmark $i$ and scenario $s$. Monthly returns are then compounded according to the horizon partition (see Fig. 5.1) following a prespecified scenario tree structure. The return scenarios in tree form are then passed on to an algebraic language deterministic model generator to produce the stochastic program deterministic equivalent instance (Consigli and Dempster, 1998).

Equity dividends are determined independently in terms of the equity position at the beginning of a stage as $\sum_{i \in l_{2}} x_{i, f_{j-1}}(s) \delta_{i, t_{j}}(s)$, where $\delta_{i, t_{j}}(s) \sim N(0.02,0.005)$ is the dividend yield.

Interest margin $M_{t_{j}}(s)=I_{t_{j}}^{+}(s)-I_{t_{j}}^{-}(s)$ is computed by subtracting negative from positive interest cash flows. Negative interest $I_{t_{j}}^{-}(s)=z_{t_{j-1}}^{-}(s) \zeta_{t_{j}}^{-}(s)$ is generated by short positions on the current account according to a fixed $2 \%$ spread over the Euribor 3 -month rate for the current period. The positive interest rate for cash surplus is fixed at $\zeta^{+}(t, s)=0.5 \%$ for all $t$ and $s$.

Positive interest $I_{t_{j}}^{+}(s)$ is calculated from return scenarios of fixed income investments $i \in I_{2}$ using buying-selling price differences assuming initial unit investments; this may be regarded as a suitable first approximation. In the case of negative price differences, the loss is entirely attributed to price movements and the interest accrual is set to 0 .
We recognize that this is a significant simplifying assumption. It is adopted here to avoid the need for an explicit yield curve model, whose introduction would go beyond the scope of a case study. Nevertheless we believe that this simplification allows one to recognize the advantages of a dynamic version of the portfolio management problem.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Liability Scenarios

Annual premium renewals by P\&C policyholders represent the key technical income on which the company relies for its ongoing business requirements. Premiums have associated random insurance claims over the years and, depending on the business cycle and the claim class, they might induce different reserve requirements. The latter constitute the insurance company’s most relevant liability. In our analysis premiums are assumed to be written annually and will be stationary over the 10-year horizon with limited volatility. Insurance claims are also assumed to remain stable over time in nominal terms, but with an inflation adjustment that in the long term may affect the company’s short-term liquidity. For a given estimate of insurance premiums $R_{0}$ collected in the first year, along scenario $s$, for $t=t_{1}, t_{2}, \ldots, H$, with $e_{r} \sim N(1,0.03)$, we assume that
$$R_{t j}(s)=\left[R_{t j-1}(s) \cdot e_{r}\right] .$$
Insurance claims are assumed to grow annually at the inflation rate and in nominal terms are constant in expectation. For given initial liability $L_{0}$, with $e_{l} \sim$ $N(1,0.01)$
$$L_{t_{j}}(s)=\left(L_{t_{j-1}}(s) \cdot e t\right)\left(1+\pi_{t_{j}}(s)\right) .$$
Every year the liability stream is assumed to vary in the following year according to a normal distribution with the previous year’s mean and a $1 \%$ volatility per year with a further inflation adjustment. Given $\pi(0)$ at time 0 , the inflation process $\pi_{t}(s)$ is assumed to follow a mean-reverting process

$$\Delta \pi_{t}(s)=\alpha_{\pi}\left(\mu-\pi_{t}(s)\right) \Delta t+\sigma_{\pi} \Delta W_{t}(s)$$
with $\mu$ set in our case study to the $2 \%$ European central bank target and $\Delta W_{t} \cdots$ $N(0, \Delta t)$. As for the other liability variables technical resenves are computed as a linear function of the current liability as $\Lambda_{t}(s)=L_{t}(s) \cdot \lambda$, where $\lambda$ in our case study is approximated by $1 / 0.3$ as estimated by practitioner opinion.

Operational costs $C_{t_{j}}(s)$ include staff and back-office and are assumed to increase at the inflation rate. For a given initial estimate $C_{0}$, along each scenario and stage, we have
$$C_{t_{j}}(s)=C_{t_{j-1}}(s) \cdot\left(1+\pi_{t_{j}}(s)\right)$$
Bad liability scenarios will be induced by years of premium reductions associated with unexpected insurance claim increases, leading to higher reserve requirements that, in turn, would require a higher capital base. A stressed situation is discussed in Section $5.4$ to emphasize the flexibility of dynamic strategies to adapt to bad technical scenarios and compensate for such losses. The application also shows that within a dynamic framework the insurance manager is able to achieve an optimal trade-off between investment and technical profit generation across scenarios and over time, and between risky and less risky portfolio positions.

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Scenario Generation

5 财产险和意外险的动态投资组合管理
109

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Asset Return Scenarios

ρ一世,吨(s): 资产回报一世有时吨在情景中s;
d一世,吨(s): 资产的股息收益率一世有时吨在情景中s;

C一世,H,吨(s)：当时的资本收益吨在情景中s单位资产投资一世∈一世2在时代H;
G吨+/−(s):现金账户当时的（正和负）利率吨在情景中s.

ρ一世,吨(s)=五一世,吨(s)−五一世,吨−Δ吨(s)五一世,吨−Δ吨(s), Δ五一世,吨(s)五一世,吨(s)=μΔ吨+ΣΔ在吨(s).

## 统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Liability Scenarios

R吨j(s)=[R吨j−1(s)⋅和r].

C吨j(s)=C吨j−1(s)⋅(1+圆周率吨j(s))

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